Differentiation of Vegetative Cells into Spores: a Kinetic ModelApplied to Bacillus subtilis

Emilie Gauvry,a Anne-Gabrielle Mathot,a Olivier Couvert,a Ivan Leguérinel,a Matthieu Jules,b Louis Corollera

aUniv Brest, Laboratoire Universitaire de Biodiversité et Ecologie Microbienne, UMT ALTER’iX, Quimper, FrancebMicalis Institute, INRA, AgroParisTech, Université Paris-Saclay, Jouy-en-Josas, France

ABSTRACT Spore-forming bacteria are natural contaminants of food raw materials,and sporulation can occur in many environments from farm to fork. In order to char-acterize and to predict spore formation over time, we developed a model that de-scribes both the kinetics of growth and the differentiation of vegetative cells intospores. The model is based on a classical growth model and enables description ofthe kinetics of sporulation with the addition of three parameters specific to sporula-tion. Two parameters are related to the probability of each vegetative cell to com-mit to sporulation and to form a spore, and the last one is related to the timeneeded to form a spore once the cell is committed to sporulation. The goodness offit of this growth-sporulation model was assessed using growth-sporulation kineticsat various temperatures in laboratory medium or in whey for Bacillus subtilis, Bacilluscereus, and Bacillus licheniformis. The model accurately describes the kinetics in thesedifferent conditions, with a mean error lower than 0.78 log10 CFU/ml for the growthand 1.08 log10 CFU/ml for the sporulation. The biological meaning of the parameterswas validated with a derivative strain of Bacillus subtilis 168 which produces greenfluorescent protein at the initiation of sporulation. This model provides physiologicalinformation on the spore formation and on the temporal abilities of vegetative cellsto differentiate into spores and reveals the heterogeneity of spore formation duringand after growth.

IMPORTANCE The growth-sporulation model describes the progressive transitionfrom vegetative cells to spores with sporulation parameters describing the sporula-tion potential of each vegetative cell. Consequently, the model constitutes an inter-esting tool to assess the sporulation potential of a bacterial population over timewith accurate parameters such as the time needed to obtain one resistant spore andthe probability of sporulation. Further, this model can be used to assess these dataunder various environmental conditions in order to better identify the conditions fa-vorable for sporulation regarding the time to obtain the first spore and/or the con-centrations of spores which could be reached during a food process.

KEYWORDS Spore-forming bacteria, cellular heterogeneity, growth modeling,sporulation

Spore-forming bacteria are common contaminants of food, and represent a majorsource of food poisoning and food spoilage (1, 2). Bacterial cells in their vegetative

or sporulated forms can be found in the environment and therefore can be naturalcontaminants of raw materials. The sporeformers display many physiological andenzymatic capacities. The spores are metabolically inactive, but they are commonlyresistant to physical and chemical treatments applied in the food industry. In contrast,vegetative cells are more sensitive to stress, are physiologically active, and can producedegradative enzymes or toxins, form biofilms, and differentiate into resistant spores.

The aim for industrials is to limit or even avoid the presence of sporeformers in their

Citation Gauvry E, Mathot A-G, Couvert O,Leguérinel I, Jules M, Coroller L. 2019.Differentiation of vegetative cells into spores: akinetic model applied to Bacillus subtilis. ApplEnviron Microbiol 85:e00322-19. https://doi.org/10.1128/AEM.00322-19.

Editor Hideaki Nojiri, University of Tokyo

Copyright © 2019 American Society forMicrobiology. All Rights Reserved.

Address correspondence to Louis Coroller,louis.coroller@univ-brest.fr.

Received 7 February 2019Accepted 10 March 2019

Accepted manuscript posted online 22March 2019Published

FOOD MICROBIOLOGY

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final products and on food lines. To do so, curative means can be used, such as cleaningand whitening processes on raw materials or physical treatments, in order to reduce theload of sporeformers. Preventive means are also used in order to limit the bacterialgrowth by applying the cold chain or to limit the formation of biofilms with cleaningand disinfection steps. However, little attention has been paid to the formation ofspores during food processes, on food lines, whereas sporulation has been observed inmilk powder processes (3, 4). The sporulation leads to an increase of the spore yield infoods. Moreover, the sporulation conditions affect the quantity and the properties ofresistance of spores to subsequent chemical or thermal treatments (5, 6). A betterunderstanding of the favorable conditions (time and physicochemical properties) forsporulation can help target the location of sporulation in the plant environment and onfood process lines in order to better prevent a high level and/or high resistance ofspores.

The modeling of microbial responses (growth, growth/no-growth, germination, orinactivation) is useful for researchers in microbiology to quantify the microbial behaviorand to test physiological hypotheses and for industrials to simulate the behavior ofpathogenic or spoilage microorganisms in order to validate the process, formulation, orconditions of storage for food (7, 8). The prediction of a microbial response could bedivided into two steps: (i) modeling the evolution of the microbial response over timewith a kinetic model and (ii) modeling the effects of environmental factors on thekinetic parameters with a secondary model. For example, bacterial growth can bepredicted over time and according to environmental factors (9–11), and some modelsexist to predict the resistance of spores to chemical and physical treatments (12–15).

Mechanistic or knowledge-based models for sporulation kinetics have been pro-posed to describe the decision-making process of sporulation initiation at the cellularand molecular levels in response to environmental stimuli (16–18). These models arecomplex because they require numerous parameters, most of which cannot be exper-imentally evaluated under industrially relevant conditions. Alternatively, empirical orphenomenological models of sporulation were proposed to describe the evolution ofspore counts over time, as they are simpler to use than mechanistic models. However,empirical models do not take into account the fact that sporulation is a process ofdifferentiation of vegetative cells into spores (19, 20), while growth and sporulation arewell known to be interdependent physiological processes (21). As far as the sporulationis concerned, no model for predicting sporulation according to environmental factorsis available in the literature yet and the kinetic models of sporulation have not attractedmuch attention in predictive microbiology.

The objectives of this work were to develop a kinetic model of sporulation which canbe used as an efficient tool to assess the sporulation under laboratory conditions and,ultimately, under food and agro-industrial conditions. The model should meet somecriteria based on the knowledge available on the sporulation process. At first, thesporulation kinetics must be deduced from the vegetative cell differentiation intospores. This implies that our model encompasses the kinetics of growth and sporula-tion. Moreover, the model parameters should characterize the heterogeneity of thetime at which the vegetative cells commit to sporulation. The model should assessthe sporulation efficiency and the time needed for the vegetative cells to complete thesporulation process. The biological meaning of the parameters linked to sporulationinitiation is assessed by the model of the Gram-positive bacterium Bacillus subtilis incombination with a fluorescent reporter of sporulation initiation (PspoIIAA gfp).

RESULTSModel development and experimental strategy. The growth of vegetative cells

was described by a modified logistic model (equation 1) and their differentiation intospores over time with specific sporulation parameters: the probability (tmax, �, and Pmax;see below) of vegetative cells to give a mature spore (heat resistant) during theincubation and the time needed for the spore formation (tf). Equation 1 is as follows:

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ln �N�ti�� � �ln �N0�, ti � �

ln �Nmax

1 � �Nmax

N0 � exp ��max � �ti � ���, ti � (1)

with N(ti) the bacterial concentration (CFU per milliliter) of the suspension at the timeti, N0 the concentration of the inoculum (CFU per milliliter), � the lag before growth (h),max the maximum vegetative growth rate (per hour), and Nmax the maximal concen-tration of total cells (CFU per milliliter). Nmax corresponds to the maximal concentrationof vegetative cells reached at the stationary phase. Once the first spores appear, Nmax

corresponds to the total cells, i.e., the spores and the remaining vegetative cells thathave not differentiated into spores.

We assumed that all vegetative cells need the same time to form a spore. Theprobability to sporulate was defined at the population level by the proportion ofvegetative cells which initiate the sporulation over time and give a mature spore (heatresistant). At the cell level, this proportion accounts for the probability of each indi-vidual cell to sporulate over time. This probability to sporulate evolves over timefollowing a normal distribution (equation 2), which is described with three parameters(Fig. 1a, c, and e). The first parameter is the maximal probability to sporulate (Pmax),which accounts for the maximal proportion of vegetative cells that can sporulate in agiven period of time. This parameter mainly influences the maximal concentration ofspores to be produced. The second one is the time (tmax) at which this maximalprobability to sporulate is obtained, which has an impact on the time at which the firstspores appear. The third parameter is the probability scattering (�), which has animpact on the speed of appearance of spores over time. Equation 2 is as follows:

P�ti� � Pmax � � 1

� � �2�� exp �0.5 � � ti � tmax

� � �2 2�� (2)

with P�ti� the probability of forming a spore at time ti (per hour) and Pmax is the maximalproportion of vegetative cells forming spores (unitless). Pmax was obtained at the timetmax (hours) at which the cell has the maximal probability of initiating sporulation and� the standard deviation around tmax (hours). Let us note that the maximal probability

FIG 1 Fluorescence and probability kinetics (a, c, and e) and growth and sporulation kinetics (b, d, and f) of B. subtilis at 27°C (a andb), 40°C (c and d), and 49°C (e and f). The values of fluorescence (diamonds) were fitted with the normal density function (solid graylines in a, c and e) and the corresponding probability densities (black dashed lines in a, c and e) with the three sporulation parametersof equation 4: Pmax, tmax, and �. The concentration of total cells (circles) and the concentration of spores (squares) over time were fittedwith the growth and sporulation model in equations 1, 2, and 3 (b, d, and f). The time between the fluorescence curve (log10 AU onthe right scale in b, d, and f) and the sporulation curve (log10 CFU/ml on the left scale in b, d, and f) corresponded to the time to forma spore tf (indicated on the time scale) fitted with equation 6.

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to sporulate at time tmax [P�tmax�] can be calculated as follows: P�tmax� � Pmax �1

� � �2�.

The experimental strategy developed to validate the biological meaning of the

sporulation parameters was based on the knowledge of the sporulation process at thephysiological level. Sporulation occurs following different signals, such as nutrientstarvation and communication molecules of quorum sensing, that require previousbacterial growth. After signal sensing (22), the sporulation starts with the activation (byphosphorylation) of the master regulator Spo0A to a given threshold of Spo0A�P.Once this threshold is reached, the activated master regulator activates the earlysporulation genes, such as spoIIAA, in the predivisional cell and triggers the asymmetricdivision to form the mother cell and the forespore (23). The sporulation processcontinues according to a sequential process involving different transcription factorsspecific to the mother cell (�E and �K) and to the forespore (�F and �G) until theformation of a mature spore. Consequently, the strategy consisted of using a promoterfusion between the gfp gene and the promoter of the gene spoIIAA (PspoIIAA gfp) as areporter of the initiation of sporulation. As the sporulation is mainly described for B.subtilis and because genetic tools are easily available for this strain, the bacterial modelB. subtilis BSB1 was used to develop the model.

We considered that within the population, each cell of the PspoIIAA gfp strain thatcommits to sporulation produces the same amount of green fluorescent protein (GFP),i.e., has the same fluorescence intensity. A sporulating cell is composed of a mother celland a forespore. The mature spore is released into the medium after lysis of the mothercell. Consequently, the fluorescence measured in a bacterial population corresponds tothe fluorescence emitted by sporulating cells in addition to the fluorescence of themedium linked to the GFP molecules released in the medium following the lysis of themother cell. The accumulation of fluorescence was directly related to the accumulationof cells that have initiated the sporulation at the time of observation and to theaccumulation of spores after the time (tf) needed to form a spore (Fig. 1b, d, and f). Tosimplify the model, we neglected the fluorescence that would be related to thepresence of GFP molecules in the refractive spores. The fluorescence (the cell commit-ment to sporulation) and the concentration of sporulating cells evolved following anormal distribution function (equation 3 and Fig. 1); consequently, the probability toform a spore over time following was assessed by a normal density function (equation3 and Fig. 1a, c, and e). Equation 3 is as follows:

F�ti� � Fmax �1

2� 1 � erf� ti � tmax

� � �2 � (3)

with F�ti� the fluorescence at time ti (arbitrary units [AU]), Fmax the maximal fluores-cence (AU), tmax (hours) the time at which Fmax (AU) is obtained, � the standarddeviation around tmax, and erf, the error function of Gauss.

Last, the time to form a spore was assessed as time needed between cell commit-ment to sporulation (increase of fluorescence) and the formation of the mature spores(heat-resistant cells) (equation 4 and Fig. 1b, d, and f). Equation 4 is as follows:

P�ti� � Pmax � N�ti� �1

2� 1 � erf� ti � tmax � tf

� � �2 � (4)

with N�ti� the concentration of total cells (equation 1), tmax (hours) the time at whichFmax (AU) was obtained, Pmax the maximal proportion of sporulating cells, and � (hours)the standard deviation around tmax (h). In this step, the values of � and tmax were setto the values determined in the previous step, by fitting the fluorescence kinetics inequation 3. The two parameters fitted on the experimental sporulation kinetics werePmax, and the time to form a spore tf.

Assessment of the growth and sporulation parameters of B. subtilis PspoIIAA gfpin modified Luria-Bertani broth at 27°C, 40°C, and 49°C. The proposed models(equations 1, 2, and 5) accurately described the growth and sporulation kinetics for B.

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subtilis in modified Luria-Bertani broth whatever the temperature of incubation. Equa-tion 5 is as follows:

S�ti� � � 0, ti � tf

S�ti � 1� � ��N�ti � tf� � S�ti � 1�� � P�ti � tf��, ti � tf

(5)

where S�ti� is spore concentration (CFU per milliliter) of the suspension at the time (ti).It can be deduced from the number of spores already formed S�ti�1� at time �ti�1� andnewly formed spores. These spores which are resistant at the time (ti) are deduced fromthe time (tf) needed from the commitment in sporulation to the completion of thespore (heat-resistant spores), the probability of the vegetative cells P�ti � tf� to committo sporulation at time �ti � tf�, and the concentration of vegetative cells which have notalready committed to sporulation. This concentration of vegetative cells corresponds tothe difference between the total cells N�ti � tf� at time �ti � tf� and the cells that arealready differentiated into spore or committed in sporulation at the time �ti � tf�. Thesetwo subpopulations correspond to the all formed spores at (ti).

The qualities of fit for growth and sporulation models reached a global root meansquare error (RMSE) of 0.77 ln CFU/ml for all tested conditions (Fig. 1). The growthkinetics were slightly better fitted, with an RMSE at 0.30 ln CFU/ml, whereas the RMSElinked to the sporulation kinetics was 0.90 ln CFU/ml with a median value at 0.60 lnCFU/ml. This loss of quality of fit is due to outliers in the spore concentration and anestimation which is more uncertain. The growth and sporulation kinetics were notsignificantly different between the wild-type BSB1 strain and the PspoIIAA gfp strain forthe three temperatures tested (likelihood ratio test, � 5%). This allowed the wild-typestrain to be used as a background to compute the fluorescence related to theproduction of GFP by the PspoIIAA gfp strain.

At 27°C, the growth rate was reduced by 35% compared to growth rate at 40°C, andthe lag time was 2-fold longer, with � values of 3.1 h and 1.6 h at 27°C and 40°Crespectively. However, the kinetics of spore appearance at 27°C and 40°C were notsignificantly different (Table 1).

The most striking differences of growth and sporulation were observed between27°C and 49°C. At 49°C, the growth was faster than at 27°C, with a rate almost 3 timeshigher and a lag before growth almost 3 times lower (Table 1). However, the sporula-tion was strongly inhibited at 49°C compared to that at 27°C (Fig. 1b and f). At 27°C, thefluorescence evolved gradually, from 0 h to 70 h, following a normal distributionfunction with a standard deviation � of 15.9 h. The time at which the fluorescenceincreased the most rapidly, i.e., when the maximal probability to sporulate (Pmax) wasobtained, was at 40 h of culture. At 49°C, the fluorescence evolved faster, with a lowerstandard deviation (6.8 h), and the maximal probability to sporulate was obtained25.1 h sooner (Fig. 1a and e). This less scattered probability of commitment to sporu-lation led to a more abrupt appearance of spores over time at 49°C than at 27°C (Fig.1b and f). Moreover, the fast appearance of spores at 49°C was increased by a short timeneeded to form a heat-resistant spore (tf) (Fig. 1b and f and Table 1).

However, fewer vegetative cells committed to sporulation at 49°C, as the maximalfluorescence reached 9.79 � 103 AU, compared to 4.11 � 104 AU at 27°C. This led to alower concentration of spores at 49°C than at 27°C (Fig. 1b and f). Indeed, the maximalconcentration of spores was 10,000 times lower. This decrease of the efficiency ofsporulation could be related to the combined evolution of the three parameters (tmax,

�, and Pmax) which define the probability to sporulate. The maximal probability (Pmax)to sporulate was almost 1,000 times lower at 49°C than at 27°C. In addition, because theprobability scattering (�) was lower at 49°C, cells were able to commit to sporulationin a shorter time frame, leading to fewer cells that were able to sporulate (Fig. 1a ande and Table 1).

Application of the growth-sporulation model in whey medium and for otherbacterial species. The growth and sporulation model (equations 1, 2, and 5) was usedto fit the kinetics of B. subtilis BSB1 cultivated in whey and of Bacillus licheniformis

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Ad978 cultivated in batch in Hageman medium supplemented with 1% brain heartinfusion (BHI) at two temperatures. Kinetics from the literature (24) were used also tofit the model, like the kinetics obtained with Bacillus cereus AH187 cultivated in thechemically defined MODS medium (50) in aerobiosis at 37°C (Fig. 2). Because of a lackof quantitative information on the time to complete the sporulation process accordingto the environmental conditions and according to the species, the time to form a spore(tf) was set to a constant value of 7 h in all cases. Let us note that such a hypothesis hadan impact on the estimates of the time (tmax) at which the maximal probability tosporulate was reached.

The model described well the kinetics of growth and sporulation, with an errorwhich was the same for both parts of the model (growth and sporulation) and whateverthe strain or the medium was. For Bacillus subtilis in whey, the root mean square errorswere 0.59 for growth and 0.86 for sporulation. The higher value for the sporulation partis explained by a higher variability in the observation and some outliers; i.e., the medianvalue of error was 0.46. For Bacillus licheniformis in Hageman medium, the root meansquare errors were 1.07 for the growth kinetics and 1.72 for the sporulation kinetics,with a median value at 1.12. Last, for Bacillus cereus in MODS medium, the root meansquare errors were 0.29 for the growth part and 0.18 for the sporulation part.

The growth-sporulation model enabled characterization of the sporulation effi-ciency and different physiological strategies used by the strains. Under conditionsunfavorable for growth, there were fewer formed spores, the mean time of commit-ment to sporulation was delayed, and its heterogeneity increased. This rising hetero-geneity was greater for Bacillus subtilis than for Bacillus licheniformis kinetics, leading toearlier or later appearance of spores than under the optimal condition of growth.

DISCUSSIONTheories and design of the model. The aim of this work was to develop a model

that describes accurately both the growth kinetics and the sporulation kinetics. Based

TABLE 1 Estimations of the fluorescence, the growth, and the sporulation parameters of B. subtilis at 27°C, 40°C, and 49°C

Data type and parameter Meaning

Estimate (95% confidence interval)

27°C 40°C 49°C

Total countN0 [ln (CFU/ml)] Initial concentration of

vegetative cells: inoculumsize

10.2 (9.6–10.7) 13.1 (12.4–13.8) 11.5 (11.1–11.9)

� (h) Lag time before growth 3.1 (2.2–3.9) 1.6 (1.1–2.1) 1.2 (0.9–1.4)max (h�1) Maximal growth rate 1.05 (0.88–1.22) 1.61 (1.33–1.88) 2.90 (2.48–3.32)Nmax [ln (CFU/ml)] Maximal concentration of total

cells20.1 (19.7–20.4) 20.0 (19.8–20.2) 19.1 (18.8–19.4)

Green fluorescenceFmax (AU) Maximal fluorescence of the

bacterial suspension(485/535 nm)

4.11 � 104 (3.85 � 104

to 4.35 � 104)5.13 � 104 (4.79 � 104

to 5.47 � 104)9.79 � 103 (7.39 � 103

to 1.22 � 104)

Heat-resistant cells (spores)tmax (h) Time at which the maximal

probability to commit tosporulation is reached

15.9 (12.5–19.4) 10.4 (5.1–15.7) 6.8 (3.3–17.0)

� (h) Standard deviation around tmax 40.0 (37.2–42.8) 36.7 (33.1–40.3) 11.6 (3.0–20.2)Pmax Maximal proportion of

sporulating vegetative cells8.86 � 10�4

(4.30 � 10�4 to1.43 � 10�3)

2.42 � 10�3

(9.14 � 10�4 to3.03 � 10�3)

4.25 � 10�7

(1.01 � 10�7 to7.51 � 10�7)

P(tmax) (h�1) Maximal probability tosporulate

2.22 � 10�5 5.44 � 10�5 2.49 � 10�8

tf (h) Time to form a spore fromcommitment to theformation of a heat-resistantspore

7.4 (7.4–7.4) 7.0 (7.0–7.0) 4.1 (4.0–4.3)

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on a classical growth model, the sporulation kinetics was precisely described usingthree parameters related to the decision-making process of vegetative cells to sporu-late and the time needed to complete the process.

The logistic model of growth (equation 1) is largely used to describe thebacterial growth. It describes the growth kinetics with the initial concentration ofcells, the lag before growth, the growth rate and the maximal concentration of totalcells. Similarly, some models were developed to describe the sporulation kineticswith parameters such as the lag before the appearance of the first spores, thesporulation rate, and the maximal concentration of spores. However, these modelsdissociate the growth and the sporulation, whereas these two bacterial processesare physiologically intertwined (25). This statement was supported by previous

FIG 2 Growth and sporulation kinetics of three Bacillus species. B. subtilis BSB1 was grown in whey medium at 25°C, 37°C, and 48°C.B. licheniformis Ad978 was grown in Hageman medium supplemented with 1% BHI at 45°C and at 20°C and pH 7.2. B. cereus AH187was grown in MODS medium in aerobiosis (24). The kinetics of total cells (circles) and the spores were fitted (solid lines) with thegrowth (equation 1) and sporulation model (equations 2 and 3). The probability to sporulate over time (equation 2) is indicatedin gray dashed lines.

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observations on other species of Bacillus, as a correlation between the growth rateand the sporulation rate was found (19).

The decision-making process to sporulate was defined elsewhere at the cell level(26–28) and was translated at the population level by the probability to sporulate (P).The sporulation decision-making process of vegetative cells is directly linked to boththe growth rate and the bacterial density (25), which evolve themselves over timefollowing the growth kinetic. Therefore, we suggested that the probability of sporula-tion evolves over time also. This hypothesis is supported by recent works (29) whichshowed that the time of sporulation (or the time at which the cells enter intosporulation) is heterogeneous among a bacterial population. For many biologicalprocesses, heterogeneity is the result of the multiscale organization of life, as explainedelsewhere (30). The heterogeneity of sporulation between cells can be explained at themolecular and cellular levels by stochastic variations (31). The heterogeneity of sporu-lation over time can be explained because the heterogeneity depends on nutrientstarvation, which becomes increasingly severe over time, and depends on quorumsensing molecules that accumulate over time. Moreover, the sporulation heterogeneityalso rises with the heterogeneity of other decision-making cell processes, such as entryinto competence, cannibalism, or dormancy (32, 33), that delay the entry into sporu-lation. Ultimately, once the sporulation is initiated by vegetative cells, the process takessome hours to achieve until it forms a mature spore, which defines the sporulationparameter (tf).

Quantitative and qualitative information brought by the sporulation parame-ters. The growth-sporulation model allowed accurate description of the growth andsporulation kinetics and allowed computation of the time to obtain the first spore in theculture, the speed of appearance of spores, and the maximal concentration of spores.Various curves shapes of growth and sporulation kinetics (fast and low kinetics)obtained for three Bacillus species and under different environmental conditions wereaccurately described with the model (Fig. 1 and 2). The model was even more accuratethan sporulation models previously suggested in the literature (19, 20) with lower RMSEvalues. In particular, these early models did not succeed in describing the smoothemergence of spores as observed at 40°C and 27°C. In some cases, the use of these earlymodels led to aberrant estimations of the time needed to see the first spores and themaximal concentration of spores.

The study of the sporulation parameters gives information on the sporulationbehavior of vegetative cells over time at the physiological level. The probability tosporulate over time is described with a normal density function involving threeparameters. The maximal probability (Pmax) to sporulate accounts for the sporulationefficiency and explains why the sporulation yield is much higher at 40°C and 27°C thanat 49°C. The low proportions of cells which sporulated at 49°C may be the result of therapid physicochemical degradation of the medium provoked by such a high temper-ature. A simple hypothesis is that the deterioration of the growth medium may alter thecell decision-making and consequently advantage or disadvantage certain physiolog-ical processes. For B. subtilis cultivated in Luria-Bertani broth at 49°C, this hypothesis issupported by the rapid cell decline observed at 49°C after the maximal concentrationof spores was reached (Fig. 2).

The probability scattering (�) assesses how synchronous the bacterial population isfor initiating sporulation. For B. subtilis, the sporulation was more synchronous at 40°Cthan at 49°C in laboratory medium (see � values in Table 1). Similarly, the species B.licheniformis, genetically close to Bacillus subtilis, displays the same characteristics, asthe sporulation is less synchronous at 45°C than at 20°C (see Table 2 and Fig. 2). At leasttwo hypotheses can explain this observation. First, the temperature affects the mem-brane fluidity by modifying its composition in fatty acids, which, in turn, is known toaffect the activity of the sensors such as the histidine kinase KinA (34). Second,differentiation processes such as the entry into competence or the cannibalism areimpacted by environmental factors. For instance, B. subtilis displays cannibalistic be-havior at 40°C but not at 45°C (35). Consequently, we can reasonably assume that there

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are fewer differentiation opportunities at 49°C than at 40°C, which leads to a lowersporulating population heterogeneity at 49°C.

Associated with � value, the time (tmax) at which Pmax is obtained allows assessmentof the time at which the first cells initiate the sporulation. It could be assessed by theproduct of the probability to sporulate with the number of total cells for a given volumeof suspension. The time needed to observe the first spores be could be estimated forvarious size of volumes of analyte (1 to 10 ml) or various sizes of batches.

Last, the time to form a spore (tf) brings information on the time needed tocomplete the sporulation process according to environmental conditions. As growthand sporulation share enzymatic machineries (36–38), the time to form a spore is likelyto be correlated with the growth rate. This could explain why the sporulation com-pleted faster at 49°C, at which bacterial cells grew faster than at 40°C and 27°C.Nevertheless, dedicated experiments are required to address this issue.

In summary, a kinetic model was developed to describe both growth and sporula-tion as a process of differentiation from vegetative cells into spores. On the one hand,the model describes the growth with the classical logistic model of Kono modified byRosso et al. (39). On the other hand, the sporulation kinetics can be describedindependently of the growth kinetics by two specific parameters: the time to form aspore and the probability to form a spore over time. The biological meaning of thesporulation parameters was experimentally assessed, providing both qualitative infor-mation at the physiological level on the sporulation process and quantitative informa-tion. The sporulation parameters revealed that at suboptimal sporulation temperatures(e.g., 49°C), vegetative cells commit to sporulation more synchronously, in smalleramounts and belatedly compared to the case at optimal temperature (e.g., 40°C). Dataof interest such as the time necessary for the formation of the first spore or the maximalconcentration of spores can be calculated. The model allows estimation of these valuesmore accurately than previous models in the literature. The model developed in thisstudy thus constitutes a good tool to describe and assess the effects of diverseenvironmental conditions on the growth and sporulation behaviors. Ultimately, thesestudies will allow identification of the favorable conditions for sporulation regardingeither the speed of the process (assessed with the time to see the first spore) and/or theconcentration of spores which could be produced (assessed with the maximal concen-tration of spores).

MATERIALS AND METHODSBiological material and strain storage. The prototrophic B. subtilis strain BSB1, a trp� derivative of

B. subtilis strain 168, was used in this work (40, 41). The BSB1 derivative strain carrying the PspoIIAA gfptranscriptional fusion was built by transformation of genomic DNA from strain AC699 (kindly providedby Arnaud Chastanet, Micalis Institute, Jouy-en-Josas, France) using natural competence. Strain AC699 isa derivative of the B. subtilis strain PY79 (42) containing the gfpmut2 gene under the control of thespoIIAA promoter (amyE::PspoIIAA gfp cat), which is a marker of the early stage of sporulation and controlsthe initiation of sporulation. The transcription of this gene is not subject to intrinsic noise, which meansthat the heterogeneity of activation of this gene is not due to stochastic processes but is correlated tothe sensing of the environment (43). GFPmut2 is stable for 7 days and in a pH range of 5.0 to 10.0 (44–46).

Regarding the transformation procedure, B. subtilis was grown overnight on Luria-Bertani plates(Difco, Becton, Dickinson and Company) at 37°C. After incubation, a colony was resuspended in MG1medium composed of MG medium containing 2 g/liter of (NH4)2SO4 (Merck, Germany), 1 g/liter ofNa3C6H5O7 (Carlo Erba Reagents, France), 14 g/liter of K2HPO4·3H2O (Sigma-Aldrich, USA), 6 g/liter ofKH2PO4 (Sigma-Aldrich), 0.5% glucose (Merck), and 15.6 mM MgSO4 (Merck), with 0.025% Casamino Acidsand 0.1% yeast extract (Difco, Becton, Dickinson and Company) added, for 4 h 30 min at 37°C underagitation at 200 rpm. A 10-fold dilution was then carried out in MG2 composed of MG medium to which0.012% Casamino Acids, 0.025% yeast extract, MgSO4, and Ca(NO3)2 (8 mM; Merck) had been added. Thesuspension was incubated for 1 h 30 min at 37°C under agitation at 200 rpm (47). A total of 200 l of thesuspension in MG2 was added to 0.1 l of genomic DNA extracted from strain AC699 with a High PurePCR template extraction kit (Roche Dignostics, Meylan, France) and incubated for 30 min at 37°C. Cloneswere selected on Luria-Bertani broth containing 5 g/ml of chloramphenicol after incubation for 24 h at37°C. The inability of the PspoIIAA gfp strain to degrade starch (as the reporter fusion is inserted at the amyElocus) was also verified on starch plates with iodine revelation.

Regarding the storage of B. subtilis strains, each selected colony was isolated on Luria-Bertani platesand incubated overnight at 37°C. A colony was resuspended in Luria-Bertani broth (Difco, Becton,Dickinson and Company) under agitation at 100 rpm at 37°C for 4 h. From this preculture, a 100-fold

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dilution was performed in 100 ml of Luria-Bertani broth in flasks, under the same culture conditions, for3 h. A second dilution was then performed under the same conditions. When the early stationary phasewas reached after a 5-h culture, glycerol was added to the bacterial suspension at a final concentrationof 25% (wt/wt) in cryovials. The bacterial cells in cryovials were stored at �80°C.

The strain B. licheniformis Ad978 was kindly provided by the Agro-Industrial Technical Institute ADRIA(Quimper, France) in cryovials containing vegetative cells conserved in BHI (Biokar Diagnostics, Beauvais,France) (70%) and glycerol (30%) at �20°C at an average concentration of 1.0 � 108 CFU/ml.

Conditions of inoculation and incubation. Before each inoculation, the absence of spores in theinoculum was ensured by heating at 80°C for 10 min the original suspension and plating 1 ml on agarplates. The absence of colonies on the plates after 24 h of incubation at 37°C confirmed the absence orlow level of resistant spores in the inoculum (�1 CFU/ml).

Vegetative cells of B. subtilis BSB1 and the PspoIIAA gfp strain were inoculated from the cryovials at aninitial concentration of 1,000 CFU/ml in 250-ml flasks filled with 100 ml of Luria-Bertani broth (BiokarDiagnostics, Beauvais, France) supplemented with sporulation salts (48). Theses flasks were incubatedunder agitation at 100 rpm at 40°C, which is close to the optimal growth temperature, or at twotemperatures suboptimal for growth and sporulation (27°C and 49°C). The incubation was performed indarkness to prevent excitation and degradation of the GFP produced by the PspoIIAA gfp strain.

Following the same procedure for inoculation, B. subtilis BSB1 was also incubated 250-ml flasks filledwith 100 ml of whey at 25°C, 37°C, or 48°C, with an initial inoculum of 1,000 CFU/ml. The whey wasprepared from dehydrated cheddar whey powder provided by ADRIA (Quimper, France) at a concen-tration of 65 g/liter. The solution was sterilized on filters with pores of 0.22 m (EMD Millipore Steritop;Fisher Scientific, France).

Batch cultures of B. licheniformis Ad978 were performed in bioreactor (Applikon Biotechnology,Holland) in 2 liters of Hageman medium (48) supplemented with 1% BHI, under agitation at 300 rpm andaeration at 3 liters/min, under two conditions: at 45°C and pH 7.2 and at 20°C and pH 7.2. The pH wasmaintained at a constant by addition of chlorydric acid (0.1 M) or sodium hydroxide (0.1 M).

Monitoring the kinetics of growth, sporulation, and fluorescence. The growth kinetics weremonitored by pouring 1 ml of the relevant dilution of the incubated medium (modified Luria-Bertanibroth, whey, or Hageman medium) into nutrient agar (Biokar Diagnostics, Beauvais, France). Enumerationof colonies was performed after incubation of the plates for 24 h at 37°C (ISO 7218). Sporulation wasmonitored by enumerating cells resistant to a 10-min heat treatment at 80°C. The heat treatment wasapplied to 100 l of suspension samples using 200-l capillary tubes which are sealed before thetreatment and aseptically broken after it (5).

The green fluorescence emitted by the total suspensions of the wild-type BSB1 (used as a referencefor background fluorescence) and PspoIIAA gfp strains was monitored over time. A total of 100 l of theincubated suspension obtained in shaking flasks was distributed in microplates, and measurements wereperformed with a microplate photometer (VICTOR X; Perkin Elmer) equipped with an excitation filter at485 nm and emission filter at 535 nm for green fluorescence measurement. The duration of the excitationwas 1.0 s.

The autofluorescence of wild-type strain BSB1 was used as the background fluorescence. Thewild-type and PspoIIAA gfp BSB1 strains were concomitantly cultivated. The fluorescence emitted bywild-type BSB1 was subtracted from the fluorescence emitted by the PspoIIAA gfp strain at each time pointto assess the fluorescence associated with the production of GFP, hereafter referred as the “fluorescence.”The detection threshold of the microplate was around 103 AU/ml.

Model development. The model of growth and sporulation can be divided into two modules. Thegrowth was fitted by the logistic model of Kono modified by Rosso et al. (39) (equation 1).

The sporulation kinetics were deduced from growth kinetics (equation 5).The probability to commit to sporulation was defined as the proportion of cells that commit to

sporulation over time. Previous works have shown that vegetative cells of a bacterial population do notinitiate the sporulation at the same time (29). Consequently, the probability of sporulation evolves overtime. In order to describe this evolution, the use of four density functions (the Gaussian, the Weibull, thelognormal, and the Gamma laws) was assessed by various criteria: the biological significance of theparameters of each model, the parsimonious number of parameters, and the quality of fit of the kineticswith the RMSE statistical criterion (see equation 7 below). This led us to choose the Gaussian probabilitydensity which was weighted by the maximal proportion Pmax of the vegetative cells to sporulate(equation 2). Finally, the sporulation part of the model combines equations 2 and 5.

Experimental assessment of the growth and the sporulation parameters. The growth and thesporulation parameters of the model (equations 1, 2, and 5) were estimated in a three-step procedurewith the cultures of the PspoIIAA gfp strain of B. subtilis. In the first step, the primary growth model wasfitted to the experimental counts (ln CFU per milliliter) to estimate the growth parameters (N0, �, max,and Nmax) with equation 1. In the second step, the fluorescence kinetics were fitted with the cumulativedistribution function for the normal distribution (equation 3).In the third step, the time taken to form aspore (tf) and the maximal proportion of sporulating cells (Pmax) were estimated. To do so, the sporulationcurves were fitted with the corresponding distribution function of equation 3, with the modification inequation 4.

Statistical procedures and analysis. The growth and sporulation parameters of equations 1 to 6were estimated by minimizing the error sum of squares (ESS; fmincon, Optimization Toolbox; MATLAB7.9.0; The Math-works, Natick, USA) (equation 6). Ninety-five percent confidence intervals were estimatedwith the nlparci function of the Optimization Toolbox (MATLAB 7.9.0; The Math-works, Natick, USA).Equation 6 is as follows:

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ESS � � �yi � y^i�2 (6)

with yi the experimental data for the concentration of total cells or spores (log CFU per milliliter) orfluorescence (AU) and y^i the value calculated with the model.

The goodness of fit of the model was assessed with the root mean square error (RMSE), determinedby equation 7:

RMSE � � ESS

n � p(7)

with ESS the error sum of squares calculated in equation 6, n the number of experimental data, and pthe number of parameters of the model.

The likelihood ratio test (49) was used to check that the growth and sporulation kinetics were notsignificantly different between the wild-type BSB1 and PspoIIAA gfp strains. The growth and sporulationparameters were estimated for both strains. In order to compare the quality of fit with the model withfitted parameters or inputs, the likelihood ratio (SL) was calculated by equation 8 (49):

SL � n � ln� ESSconstrained

ESSunconstrained (8)

where n is the number of experimental data, ESSunconstrained is the ESS obtained by fitting the eightgrowth and sporulation parameters to the kinetics of the PspoIIAA gfp strain, and ESSconstrained is the ESSobtained with the same eight kinetics but using the 8 parameters estimated for strain BSB1 as inputs. Thevalue was compared with the chi-squared value (15.51) that corresponds to a degree of freedom of 8 anda tolerance threshold of 5%.

ACKNOWLEDGMENTSWe thank Arnaud Chastanet (Micalis, Jouy-en-Josas, France) for providing the B.

subtilis AC699 strain.This work was supported by Quimper Bretagne Occidentale and by a doctoral grant

from the regional council of Brittany (Région Bretagne, France).We declare no conflict of interest.

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